A finite difference scheme for smooth solutions of the general mKdV equation

We consider a generalization of the mKdV model of shallow water out-flows. This generalization is a family of equations with nonlinear dispersion terms containing, in particular, KdV, mKdV, Benjamin-Bona-Mahony, Camassa-Holm, and Degasperis-Procesi equations. Nonlinear dispersion, generally speaking, implies instability of classical solutions and wave breaking in a finite time. However, there are special conditions under which the general mKdV equation admits classical solutions that are global in time. We have created an economic finite difference scheme that preserves this property for numerical solutions. To illustrate this we demonstrate some numerical results about propagation and interaction of solitons.